Abstract: A great deal of often seemingly elementary questions concerning the geometry of affine algebraic varieties are intimately linked to the central problem of how to recognize and characterize affine spaces among algebraic varieties. I will give an overview of classical results on this question using a variety of techniques, ranging from commutative algebra to differential topology to logarithmic birational geometry, and then highlight some more recent advances and new perspectives opened by ideas and techniques at the interface between algebraic geometry and $\mathbb{A}^1$ -homotopy theory.
Abstract: Whenever one studies a mathematical object one ought also to study its symmetries. Manifolds---spaces which look locally like ordinary Euclidean space but which can be globally complicated---are the central objects of study in topology and geometry, and their groups of symmetries come in many flavours (isometries, diffeomorphisms, homeomorphisms, ...). I will discuss some classical and recent results about the spaces of all symmetries of certain simple manifolds, and report on an emerging conjectural picture.